Rectangular multi-$t$ Yang–Baxter trace conjecture

Let $bc=(k^r):=(k,k,\ldots,k)$ be a rectangular partition, and let $c3_bc$ and $v_bc$ be the permutation and spectral-parameter vector associated with $bc$. The vector is

$$v_\mu=\left(1,q,\ldots,q^{k-1}, \frac{t_{r-1}}{t_{r-2}},\ldots,q^{k-1}\frac{t_{r-1}}{t_{r-2}},\frac{t_{r-1}}{t_{r-3}},\ldots,q^{k-1}\frac{t_{r-1}}{t_{r-3}},\ldots,t_{r-1},\ldots,q^{k-1}t_{r-1}\right).$$

Rectangular multi-$t$ Yang–Baxter trace conjecture. The equivariant trace of $Y_{c3_bc}(v_bc)$ is proportional to

$$\omega \tilde H_\mu((1-q)X;q,t_1,\ldots,t_{r-1}).$$

The conjecture is motivated by constructions and computations for rectangular partitions; the source does not report a general proof.